Magnus Find, Alexander Golovnev, E. Hirsch, A. Kulikov
We consider Boolean circuits over the full binary basis. We prove a (3+1/86)n-o(n) lower bound on the size of such a circuit for an explicitly defined predicate, namely an affine disperser for sublinear dimension. This improves the 3n-o(n) bound of Norbert Blum (1984).The proof is based on the gate elimination technique extended with the following three ideas. We generalize the computational model by allowing circuits to contain cycles, this in turn allows us to perform affine substitutions. We use a carefully chosen circuit complexity measure to track the progress of the gate elimination process. Finally, we use quadratic substitutions that may be viewed as delayed affine substitutions.
{"title":"A Better-Than-3n Lower Bound for the Circuit Complexity of an Explicit Function","authors":"Magnus Find, Alexander Golovnev, E. Hirsch, A. Kulikov","doi":"10.1109/FOCS.2016.19","DOIUrl":"https://doi.org/10.1109/FOCS.2016.19","url":null,"abstract":"We consider Boolean circuits over the full binary basis. We prove a (3+1/86)n-o(n) lower bound on the size of such a circuit for an explicitly defined predicate, namely an affine disperser for sublinear dimension. This improves the 3n-o(n) bound of Norbert Blum (1984).The proof is based on the gate elimination technique extended with the following three ideas. We generalize the computational model by allowing circuits to contain cycles, this in turn allows us to perform affine substitutions. We use a carefully chosen circuit complexity measure to track the progress of the gate elimination process. Finally, we use quadratic substitutions that may be viewed as delayed affine substitutions.","PeriodicalId":414001,"journal":{"name":"2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"62 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129940426","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the problem of compressing interactive communication to its information content I, defined as the amount of information that the participants learn about each other's inputs. We focus on the case when the participants' inputs are distributed independently and show how to compress the communication to O(I log2 I) bits, with no dependence on the original communication cost. This result improves quadratically on previous work by Kol (STOC 2016) and essentially matches the well-known lower bound Ω(I).
{"title":"Compressing Interactive Communication under Product Distributions","authors":"Alexander A. Sherstov","doi":"10.1109/FOCS.2016.64","DOIUrl":"https://doi.org/10.1109/FOCS.2016.64","url":null,"abstract":"We study the problem of compressing interactive communication to its information content I, defined as the amount of information that the participants learn about each other's inputs. We focus on the case when the participants' inputs are distributed independently and show how to compress the communication to O(I log2 I) bits, with no dependence on the original communication cost. This result improves quadratically on previous work by Kol (STOC 2016) and essentially matches the well-known lower bound Ω(I).","PeriodicalId":414001,"journal":{"name":"2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"882 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122069183","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In a recent breakthrough [1], Chattopadhyay and Zuckerman gave an explicit two-source extractor for min-entropy k ≥ logC n for some large enough constant C, where n is the length of the source. However, their extractor only outputs one bit. In this paper, we improve the output of the two-source extractor to kΩ(1), while the error remains n-Ω(1) and the extractor remains strong in the second source. In the non-strong case, the output can be increased to k. Our improvement is obtained by giving a better extractor for (q, t, γ) non-oblivious bit-fixing sources, which can output tΩ(1) bits instead of one bit as in [1]. We also give the first explicit construction of deterministic extractors for affine sources over F2, with entropy k ≥ logC n for some large enough constant C, where n is the length of the source. Previously the best known results are by Bourgain [2], Yehudayoff [3] and Li [4], which require the affine source to have entropy at least Ω(n/√log log n). Our extractor outputs kΩ(1) bits with error n-Ω(1). This is done by reducing an affine source to a non-oblivious bit-fixing source, where we adapt the alternating extraction based approach in previous work on independent source extractors [5] to the affine setting. Our affine extractors also imply improved extractors for circuit sources studied in [6]. We further extend our results to the case of zero-error dispersers, and give two applications in data structures that rely crucially on the fact that our two-source or affine extractors have large output size.
{"title":"Improved Two-Source Extractors, and Affine Extractors for Polylogarithmic Entropy","authors":"Xin Li","doi":"10.1109/FOCS.2016.26","DOIUrl":"https://doi.org/10.1109/FOCS.2016.26","url":null,"abstract":"In a recent breakthrough [1], Chattopadhyay and Zuckerman gave an explicit two-source extractor for min-entropy k ≥ logC n for some large enough constant C, where n is the length of the source. However, their extractor only outputs one bit. In this paper, we improve the output of the two-source extractor to kΩ(1), while the error remains n-Ω(1) and the extractor remains strong in the second source. In the non-strong case, the output can be increased to k. Our improvement is obtained by giving a better extractor for (q, t, γ) non-oblivious bit-fixing sources, which can output tΩ(1) bits instead of one bit as in [1]. We also give the first explicit construction of deterministic extractors for affine sources over F2, with entropy k ≥ logC n for some large enough constant C, where n is the length of the source. Previously the best known results are by Bourgain [2], Yehudayoff [3] and Li [4], which require the affine source to have entropy at least Ω(n/√log log n). Our extractor outputs kΩ(1) bits with error n-Ω(1). This is done by reducing an affine source to a non-oblivious bit-fixing source, where we adapt the alternating extraction based approach in previous work on independent source extractors [5] to the affine setting. Our affine extractors also imply improved extractors for circuit sources studied in [6]. We further extend our results to the case of zero-error dispersers, and give two applications in data structures that rely crucially on the fact that our two-source or affine extractors have large output size.","PeriodicalId":414001,"journal":{"name":"2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129550129","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let f be a boolean function on n variables. Its associated XOR function is the two-party function F(x, y) = f(x xor y). We show that, up to polynomial factors, the deterministic communication complexity of F is equal to the parity decision tree complexity of f. This relies on a novel technique of entropy reduction for protocols, combined with existing techniques in Fourier analysis and additive combinatorics.
{"title":"Structure of Protocols for XOR Functions","authors":"Hamed Hatami, Kaave Hosseini, Shachar Lovett","doi":"10.1109/FOCS.2016.38","DOIUrl":"https://doi.org/10.1109/FOCS.2016.38","url":null,"abstract":"Let f be a boolean function on n variables. Its associated XOR function is the two-party function F(x, y) = f(x xor y). We show that, up to polynomial factors, the deterministic communication complexity of F is equal to the parity decision tree complexity of f. This relies on a novel technique of entropy reduction for protocols, combined with existing techniques in Fourier analysis and additive combinatorics.","PeriodicalId":414001,"journal":{"name":"2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"120 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122476612","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Susanna F. de Rezende, Jakob Nordström, Marc Vinyals
We obtain the first true size-space trade-offs for the cutting planes proof system, where the upper bounds hold for size and total space for derivations with constantsize coefficients, and the lower bounds apply to length and formula space (i.e., number of inequalities in memory) even for derivations with exponentially large coefficients. These are also the first trade-offs to hold uniformly for resolution, polynomial calculus and cutting planes, thus capturing the main methods of reasoning used in current state-of-the-art SAT solvers. We prove our results by a reduction to communication lower bounds in a round-efficient version of the real communication model of [Kraj́ĩcek '98], drawing on and extending techniques in [Raz and McKenzie '99] and [G̈öos et al. '15]. The communication lower bounds are in turn established by a reduction to trade-offs between cost and number of rounds in the game of [Dymond and Tompa '85] played on directed acyclic graphs. As a by-product of the techniques developed to show these proof complexity trade-off results, we also obtain an exponential separation between monotone-ACi-1 and monotone-ACi, improving exponentially over the superpolynomial separation in [Raz and McKenzie '99]. That is, we give an explicit Boolean function that can be computed by monotone Boolean circuits of depth logi n and polynomial size, but for which circuits of depth O(logi-1 n) require exponential size.
{"title":"How Limited Interaction Hinders Real Communication (and What It Means for Proof and Circuit Complexity)","authors":"Susanna F. de Rezende, Jakob Nordström, Marc Vinyals","doi":"10.1109/FOCS.2016.40","DOIUrl":"https://doi.org/10.1109/FOCS.2016.40","url":null,"abstract":"We obtain the first true size-space trade-offs for the cutting planes proof system, where the upper bounds hold for size and total space for derivations with constantsize coefficients, and the lower bounds apply to length and formula space (i.e., number of inequalities in memory) even for derivations with exponentially large coefficients. These are also the first trade-offs to hold uniformly for resolution, polynomial calculus and cutting planes, thus capturing the main methods of reasoning used in current state-of-the-art SAT solvers. We prove our results by a reduction to communication lower bounds in a round-efficient version of the real communication model of [Kraj́ĩcek '98], drawing on and extending techniques in [Raz and McKenzie '99] and [G̈öos et al. '15]. The communication lower bounds are in turn established by a reduction to trade-offs between cost and number of rounds in the game of [Dymond and Tompa '85] played on directed acyclic graphs. As a by-product of the techniques developed to show these proof complexity trade-off results, we also obtain an exponential separation between monotone-ACi-1 and monotone-ACi, improving exponentially over the superpolynomial separation in [Raz and McKenzie '99]. That is, we give an explicit Boolean function that can be computed by monotone Boolean circuits of depth logi n and polynomial size, but for which circuits of depth O(logi-1 n) require exponential size.","PeriodicalId":414001,"journal":{"name":"2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"56 9","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134225866","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the robust curve fitting problem, for both algebraic and Fourier (trigonometric) polynomials, in the presence of outliers. In particular, we study the model of Arora and Khot (STOC 2002), who were motivated by applications in computer vision. In their model, the input data consists of ordered pairs (xi, yi) ε [-1, 1] × [-1, 1], i = 1, 2,..., N, and there is an unknown degree-d polynomial p such that for all but ρ fraction of the i, we have |p(xi) - yi|≤ δ. Unlike Arora-Khot, we also study the trigonometric setting, where the input is from T × [-1, 1], where T is the unit circle. In both scenarios, the i corresponding to errors are chosen randomly, and for such i the errors in the yi can be arbitrary. The goal is to output a degree-d polynomial q such that ||p - q||∞ is small (for example, O(δ)). Arora and Khot could achieve a polynomial-time algorithm only for ρ = 0. Daltrophe et al. observed that a simple median-based algorithm can correct errors if the desired accuracy δ is large enough. (Larger δ makes the output guarantee easier to achieve, which seems to typically outweigh the weaker input promise.) We dramatically expand the range of parameters for which recovery of q is possible in polynomial time. Specifically, we show that there are polynomial-time algorithms in both settings that recover q up to l∞ error O(δ.99) provided 1) ρ ≤/c1log d and δ ≥ 1/(log d)c, or 2) ρ ≤ c1/log log d/log2 d and δ ≥ 1/dc. Here c is any constant and c1 is a small enough constant depending on c. The number of points that suffices is N = Õ(d) in the trigonometric setting for random xi or arbitrary xi that are roughly equally spaced, or in the algebraic setting when the xi are chosen according to the Chebyshev distribution, and N = Õ(d2) in the algebraic setting with random (or roughly equally spaced) xi.
{"title":"Robust Fourier and Polynomial Curve Fitting","authors":"V. Guruswami, David Zuckerman","doi":"10.1109/FOCS.2016.75","DOIUrl":"https://doi.org/10.1109/FOCS.2016.75","url":null,"abstract":"We consider the robust curve fitting problem, for both algebraic and Fourier (trigonometric) polynomials, in the presence of outliers. In particular, we study the model of Arora and Khot (STOC 2002), who were motivated by applications in computer vision. In their model, the input data consists of ordered pairs (x<sub>i</sub>, y<sub>i</sub>) ε [-1, 1] × [-1, 1], i = 1, 2,..., N, and there is an unknown degree-d polynomial p such that for all but ρ fraction of the i, we have |p(x<sub>i</sub>) - y<sub>i</sub>|≤ δ. Unlike Arora-Khot, we also study the trigonometric setting, where the input is from T × [-1, 1], where T is the unit circle. In both scenarios, the i corresponding to errors are chosen randomly, and for such i the errors in the yi can be arbitrary. The goal is to output a degree-d polynomial q such that ||p - q||<sub>∞</sub> is small (for example, O(δ)). Arora and Khot could achieve a polynomial-time algorithm only for ρ = 0. Daltrophe et al. observed that a simple median-based algorithm can correct errors if the desired accuracy δ is large enough. (Larger δ makes the output guarantee easier to achieve, which seems to typically outweigh the weaker input promise.) We dramatically expand the range of parameters for which recovery of q is possible in polynomial time. Specifically, we show that there are polynomial-time algorithms in both settings that recover q up to l∞ error O(δ.99) provided 1) ρ ≤/c1log d and δ ≥ 1/(log d)c, or 2) ρ ≤ c1/log log d/log2 d and δ ≥ 1/dc. Here c is any constant and c1 is a small enough constant depending on c. The number of points that suffices is N = Õ(d) in the trigonometric setting for random x<sub>i</sub> or arbitrary x<sub>i</sub> that are roughly equally spaced, or in the algebraic setting when the x<sub>i</sub> are chosen according to the Chebyshev distribution, and N = Õ(d2) in the algebraic setting with random (or roughly equally spaced) x<sub>i</sub>.","PeriodicalId":414001,"journal":{"name":"2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133220338","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose definitions of substitutes and complements for pieces of information ("signals") in the context of a decision or optimization problem, with game-theoretic and algorithmic applications. In a game-theoretic context, substitutes capture diminishing marginal value of information to a rational decision maker. There, we address the main open problem in a fundamental strategic-information-revelation setting, prediction markets. We show that substitutes characterize "best-possible" equilibria with immediate information aggregation, while complements characterize "worst-possible", delayed aggregation. Game-theoretic applications also include settings such as crowdsourcing contests and question-and-answer forums. In an algorithmic context, where substitutes capture diminishing marginal improvement of information to an optimization problem, substitutes imply efficient approximation algorithms for a very general class of (adaptive) information acquisition problems. In tandem with these broad applications, we examine the structure and design of informational substitutes and complements. They have equivalent, intuitive definitions from disparate perspectives: submodularity, geometry, and information theory. We also consider the design of scoring rules or optimization problems so as to encourage substitutability or complementarity, with positive and negative results. Taken as a whole, the results give some evidence that, in parallel with substitutable items, informational substitutes play a natural conceptual and formal role in game theory and algorithms.
{"title":"Informational Substitutes","authors":"Yiling Chen, Bo Waggoner","doi":"10.1109/FOCS.2016.33","DOIUrl":"https://doi.org/10.1109/FOCS.2016.33","url":null,"abstract":"We propose definitions of substitutes and complements for pieces of information (\"signals\") in the context of a decision or optimization problem, with game-theoretic and algorithmic applications. In a game-theoretic context, substitutes capture diminishing marginal value of information to a rational decision maker. There, we address the main open problem in a fundamental strategic-information-revelation setting, prediction markets. We show that substitutes characterize \"best-possible\" equilibria with immediate information aggregation, while complements characterize \"worst-possible\", delayed aggregation. Game-theoretic applications also include settings such as crowdsourcing contests and question-and-answer forums. In an algorithmic context, where substitutes capture diminishing marginal improvement of information to an optimization problem, substitutes imply efficient approximation algorithms for a very general class of (adaptive) information acquisition problems. In tandem with these broad applications, we examine the structure and design of informational substitutes and complements. They have equivalent, intuitive definitions from disparate perspectives: submodularity, geometry, and information theory. We also consider the design of scoring rules or optimization problems so as to encourage substitutability or complementarity, with positive and negative results. Taken as a whole, the results give some evidence that, in parallel with substitutable items, informational substitutes play a natural conceptual and formal role in game theory and algorithms.","PeriodicalId":414001,"journal":{"name":"2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"55 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121659948","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We obtain a new depth-reduction construction, which implies a super-exponential improvement in the depth lower bound separating NEXP from non-uniform ACC. In particular, we show that every circuit with AND, OR, NOT, and MODm gates, m ε Z+, of polynomial size and depth d can be reduced to a depth-2, SYM-AND, circuit of size 2(log n)O(d). This is an exponential size improvement over the traditional Yao-Beigel-Tarui, which has size blowup 2(log n)2O(d). Therefore, depth-reduction for composite m matches the size of the Allender-Hertrampf construction for primes from 1989. One immediate implication of depth reduction is an improvement of the depth from o(loglog n) to o(log n/loglog n), in Williams' program for ACC circuit lower bounds against NEXP. This is just short of O(log n/loglog n) and thus pushes William's program to the NC1 barrier, since NC1 is contained in ACC of depth O(log n/loglog n). A second, but non-immediate, implication regards the strengthening of the ACC lower bound in the Chattopadhyay-Santhanam interactive compression setting.
{"title":"Depth-Reduction for Composites","authors":"Shiteng Chen, Periklis A. Papakonstantinou","doi":"10.1109/FOCS.2016.20","DOIUrl":"https://doi.org/10.1109/FOCS.2016.20","url":null,"abstract":"We obtain a new depth-reduction construction, which implies a super-exponential improvement in the depth lower bound separating NEXP from non-uniform ACC. In particular, we show that every circuit with AND, OR, NOT, and MODm gates, m ε Z+, of polynomial size and depth d can be reduced to a depth-2, SYM-AND, circuit of size 2(log n)O(d). This is an exponential size improvement over the traditional Yao-Beigel-Tarui, which has size blowup 2(log n)2O(d). Therefore, depth-reduction for composite m matches the size of the Allender-Hertrampf construction for primes from 1989. One immediate implication of depth reduction is an improvement of the depth from o(loglog n) to o(log n/loglog n), in Williams' program for ACC circuit lower bounds against NEXP. This is just short of O(log n/loglog n) and thus pushes William's program to the NC1 barrier, since NC1 is contained in ACC of depth O(log n/loglog n). A second, but non-immediate, implication regards the strengthening of the ACC lower bound in the Chattopadhyay-Santhanam interactive compression setting.","PeriodicalId":414001,"journal":{"name":"2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128788448","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The following procedure yields an unbiased estimator for the disconnection probability of an n-vertex graph with minimum cut c if every edge fails independently with probability p: (i) contract every edge independently with probability 1- n-2/c, then (ii) recursively compute the disconnection probability of the resulting tiny graph if each edge fails with probability n2/cp. We give a short, simple, self-contained proof that this estimator can be computed in linear time and has relative variance O(n2). Combining these two facts with a standard sparsification argument yields an O(n3 log n)-time algorithm for estimating the (un)reliability of a network. We also show how the technique can be used to create unbiased samples of disconnected networks.
{"title":"A Fast and Simple Unbiased Estimator for Network (Un)reliability","authors":"David R Karger","doi":"10.1109/FOCS.2016.96","DOIUrl":"https://doi.org/10.1109/FOCS.2016.96","url":null,"abstract":"The following procedure yields an unbiased estimator for the disconnection probability of an n-vertex graph with minimum cut c if every edge fails independently with probability p: (i) contract every edge independently with probability 1- n-2/c, then (ii) recursively compute the disconnection probability of the resulting tiny graph if each edge fails with probability n2/cp. We give a short, simple, self-contained proof that this estimator can be computed in linear time and has relative variance O(n2). Combining these two facts with a standard sparsification argument yields an O(n3 log n)-time algorithm for estimating the (un)reliability of a network. We also show how the technique can be used to create unbiased samples of disconnected networks.","PeriodicalId":414001,"journal":{"name":"2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123842214","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the problem of estimating a Fourier-sparse signal from noisy samples, where the sampling is done over some interval [0, T] and the frequencies can be "off-grid". Previous methods for this problem required the gap between frequencies to be above 1/T, the threshold required to robustly identify individual frequencies. We show the frequency gap is not necessary to estimate the signal as a whole: for arbitrary k-Fourier-sparse signals under l2 bounded noise, we show how to estimate the signal with a constant factor growth of the noise and sample complexity polynomial in k and logarithmic in the bandwidth and signal-to-noise ratio. As a special case, we get an algorithm to interpolate degree d polynomials from noisy measurements, using O(d) samples and increasing the noise by a constant factor in l2.
{"title":"Fourier-Sparse Interpolation without a Frequency Gap","authors":"Xue Chen, D. Kane, Eric Price, Zhao Song","doi":"10.1109/FOCS.2016.84","DOIUrl":"https://doi.org/10.1109/FOCS.2016.84","url":null,"abstract":"We consider the problem of estimating a Fourier-sparse signal from noisy samples, where the sampling is done over some interval [0, T] and the frequencies can be \"off-grid\". Previous methods for this problem required the gap between frequencies to be above 1/T, the threshold required to robustly identify individual frequencies. We show the frequency gap is not necessary to estimate the signal as a whole: for arbitrary k-Fourier-sparse signals under l2 bounded noise, we show how to estimate the signal with a constant factor growth of the noise and sample complexity polynomial in k and logarithmic in the bandwidth and signal-to-noise ratio. As a special case, we get an algorithm to interpolate degree d polynomials from noisy measurements, using O(d) samples and increasing the noise by a constant factor in l2.","PeriodicalId":414001,"journal":{"name":"2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"123 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128355450","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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